Contents - ChaosBook

CHAPTER 2. GO WITH THE FLOW 36or any combination of the above. The most interesting case is that of an aperiodicrecurrent attractor, to which we shall refer loosely as a strange attractor. We say [example 2.3]‘loosely’, as will soon become apparent that diagnosing and proving existence ofa genuine, card-carrying strange attractor is a highly nontrivial undertaking.Conversely, if we can enclose the non–wandering set Ω by a connected statespace volume M 0 and then show that almost all points within M 0 , but not inΩ, eventually exit M 0 , we refer to the non–wandering set Ω as a repeller. Anexample of a repeller is not hard to come by–the pinball game of sect. 1.3 is asimple chaotic repeller.It would seem, having said that the periodic points are so exceptional thatalmost all non-wandering points are aperiodic, that we have given up the ancients’fixation on periodic motions. Nothing could be further from truth. As longer andlonger cycles approximate more and more accurately finite segments of aperiodictrajectories, we shall establish control over non–wandering sets by defining themas the closure of the union of all periodic points.Before we can work out an example of a non–wandering set and get a bettergrip on what chaotic motion might look like, we need to ponder flows in a littlemore depth.2.2 FlowsThere is no beauty without some strangeness.—William BlakeA flow is a continuous-time dynamical system. The evolution rule f t is a familyof mappings of M→Mparameterized by t ∈ R. Because t represents a timeinterval, any family of mappings that forms an evolution rule must satisfy:[exercise 2.2](a) f 0 (x) = x(in 0 time there is no motion)(b) f t ( f t′ (x)) = f t+t′ (x)(the evolution law is the same at all times)(c) the mapping (x, t) ↦→ f t (x) from M×R into M is continuous.We shall often find it convenient to represent functional composition by ‘◦ :’[appendix H.1]f t+s = f t ◦ f s = f t ( f s ) . (2.3)The family of mappings f t (x) thus forms a continuous (forward semi-) group.Why ‘semi-’group? It may fail to form a group if the dynamics is not reversible,and the rule f t (x) cannot be used to rerun the dynamics backwards in time, withnegative t; with no reversibility, we cannot define the inverse f −t ( f t (x)) = f 0 (x) =x , in which case the family of mappings f t (x) does not form a group. In exceedinglymany situations of interest–for times beyond the Lyapunov time, forflows - 1apr2008.tex